A **combination** is the number of ways a given number of objects can be selected from a group when the order does not matter (unlike a permutation, in which order does matter). If *k* objects are selected from a group of *n* members, the formula for the combination (which can be read as "n choose k") is

- $ {n \choose k} = \frac{n!}{k! (n-k)!} $

Combinations are important in probability as well as to the binomial theorem. All the possible combinations of an integer *n* make up the *n*th row of Pascal's triangle.

## Example

How many different groups of three can be chosen from five people?

Since order is not important, we can use the formula for a combination.

- $ {5 \choose 3} = \frac{5!}{3! (5-3)!} = \frac{5!}{3! \cdot (2)!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot (2 \cdot 1)} = \frac{5 \cdot 4}{2} = 10 $

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